Estimation of formation elastic constants from drilling

ABSTRACT

Systems and methods are provided for identifying one or more properties of a rock formation. A system may be configured to receive sensor data obtained from one or more sensors, the sensor data generated from drilling operations performed in a wellbore, generate stress metrics and strain metrics for the sensor data based on a calibrated stress and strain map, and identify one or more properties of a rock formation based on the stress metrics and the strain metrics.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of, and priority to, U.S.Provisional Patent Application No. 62/861,720 filed on Jun. 14, 2019,entitled “ESTIMATION OF FORMATION ELASTIC CONSTANTS FROM DRILLING DATA,”the contents of which are hereby expressly incorporated by reference intheir entirety.

TECHNICAL FIELD

The present disclosure is related generally to identifying rockproperties, and more particularly to identifying rock properties fordrilling.

BACKGROUND

During drilling operations, operators employ any number of methods anddevices to ascertain information about downhole conditions. For example,information about a formation's lithology and the type of formationfluid can be determined using wireline logging ormeasurement-while-drilling (MWD) techniques. In developed reservoirs,similar information can be inferred from logs of offset wells,geological maps, and the like. Cuttings circulated to the surface mayalso be used to identify a formation's lithology, and changes in mudweight or resistivity (in water based muds) can be used to indicate thepresence of hydrocarbons. Numerous other techniques and tools may alsobe used to gather information about downhole conditions.

Most of the methods mentioned above, however, do not provide real-timeinformation during drilling operations. Wireline logging can beconducted only after the formation of interest has already been drilledthrough. Offset logs offer only general guidance, since there is noguarantee that the location, porosity, thickness, etc. of a formation ofinterest will be the same between offset wells and a well being drilled.Cuttings and changes in mud characteristics are known only after mudthat is at the bit has had time to circulate to the surface. Some MWDtools require drilling to be suspended while a wireline or slickline isdropped through the drill string to retrieve data recorded by thedownhole tool.

Acoustic-type MWD tools may provide information about formationproperties in real-time. Acoustic-type tools generally measure variousproperties of acoustic signals (e.g. the time it takes an acousticsignal to travel from a transmitter, through the formation, and back toa receiver) to determine properties of the rock surrounding thewellbore. Such tools may be hindered by, for example, the acousticsignal being required to pass through multiple formation types havingdifferent acoustic properties, variations in mud density, noisegenerated by the drill string, or the like.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the manner in which the above-recited and otheradvantages and features of the disclosure can be obtained, a moreparticular description of the principles briefly described above will berendered by reference to specific embodiments thereof which areillustrated in the appended drawings. Understanding that these drawingsdepict only exemplary embodiments of the disclosure and are nottherefore to be considered to be limiting of its scope, the principlesherein are described and explained with additional specificity anddetail through the use of the accompanying drawings in which:

FIG. 1A is a schematic diagram of an example Logging While Drillingwellbore operating environment, in accordance with various aspects ofthe subject technology;

FIG. 1B is a schematic diagram of an example downhole environment, postdrilling, in accordance with various aspects of the subject technology;

FIG. 2 is a flow diagram showing a top-level workflow for estimatingelastic constants from drilling data, in accordance with various aspectsof the subject technology;

FIG. 3 is a diagram illustrating a workflow for preprocessing drillingdata, in accordance with various aspects of the subject technology;

FIG. 4 is a diagram illustrating a calibration workflow, in accordancewith various aspects of the subject technology;

FIG. 5 is a diagram illustrating an adaptive linear element (ADALINE)neural network representation of the stress and strain, in accordancewith various aspects of the subject technology;

FIG. 6 is a diagram illustrating stress and strain networks trainagainst each other using the stress-strain relationship for knownstiffness matrix C, in accordance with various aspects of the subjecttechnology;

FIG. 7 is a diagram illustrating an example method of identifyingproperties of a rock formation based on stress and strain metrics, inaccordance with various aspects of the subject technology; and

FIG. 8 is a diagram illustrating an example computing devicearchitecture of a computing device which can implement the varioustechnologies and techniques described herein, in accordance with variousaspects of the subject technology.

DETAILED DESCRIPTION

Various embodiments of the disclosure are discussed in detail below.While specific implementations are discussed, it should be understoodthat this is done for illustration purposes only. A person skilled inthe relevant art will recognize that other components and configurationsmay be used without parting from the spirit and scope of the disclosure.

Additional features and advantages of the disclosure will be set forthin the description which follows, and in part will be obvious from thedescription, or can be learned by practice of the herein disclosedprinciples. The features and advantages of the disclosure can berealized and obtained by means of the instruments and combinationsparticularly pointed out in the appended claims. These and otherfeatures of the disclosure will become more fully apparent from thefollowing description and appended claims, or can be learned by thepractice of the principles set forth herein.

It will be appreciated that for simplicity and clarity of illustration,where appropriate, reference numerals have been repeated among thedifferent figures to indicate corresponding or analogous elements. Inaddition, numerous specific details are set forth in order to provide athorough understanding of the embodiments described herein. However, itwill be understood by those of ordinary skill in the art that theembodiments described herein can be practiced without these specificdetails. In other instances, methods, procedures and components have notbeen described in detail so as not to obscure the related relevantfeature being described. The drawings are not necessarily to scale andthe proportions of certain parts may be exaggerated to better illustratedetails and features. The description is not to be considered aslimiting the scope of the embodiments described herein.

Aspects of the subject technology relate to improved systems and methodsfor estimating acoustic elastic constants from drilling data inreal-time (while drilling) or near real-time based on calibration froman offset well, a library, or a depth interval of the currently drilledwell. According to some embodiments, borehole drilling sensors recorddata such as depth, time, acceleration, rate of penetration (ROP),weight on bit (WOB), torque on bit (TOB), magnetometer and gyroscopedata, and the like. Other sensors may also be on the drill stringproviding acoustic data, density measurements, gamma ray measurements,and the like. A computing system may be configured to collect the datafrom the various sources and relate the statistical properties of therecorded data to the acoustic properties of the rocks in the formation.The system may further calibrate a linear mapping of sensor data tostress and strain using independently obtained elastic constant data.The independent data may come from various sources such as wireline orlogging while drilling (LWD acoustic and density data, laboratorymeasurements based on core cuttings, or from a library developed overtime from prior calibrations for different types of drill bit.

According to various embodiments, the calibration performed by thesystem not only estimates the parameters of the linear map, but alsooptimizes the structure of the linear map. Once an optimal mapping isdetermined with regards to type of stress and/or strain excitation,number of drilling parameters used in the inversion, and number ofestimated acoustic elastic constants, multiple linear regression is usedto invert for the acoustic elastic constants as a function of depthwithout additional calibration. The multiple linear regression performedby the system is configured to model the relationship between two ormore explanatory variables and a response variable by fitting a multiplelinear equation to observed data. The resulting estimates are obtainedcheaply in real-time or near real-time without the need of additionalexpensive sonic measurements. The information can be used to designcompletions and improve perforation efficiency.

Aspects of the subject technology optimize and calibrate a generallinear mapping of sensor data to stress and strain using independentlyobtained elastic constant data. As a result, the accuracy of theregression analysis used to estimate the elastic constants is improvedover alternative techniques of estimation of rock properties fromdrilling data. A discussion of some alternative approaches and theirshortcomings are provided below. Aspects of the subject technologyaddress shortcomings of these alternative approaches and provide for animprovement over these alternative approaches.

Drilling noise data may correlate well with mechanical rock propertiesand some techniques for estimation of rock properties may use drillingnoise spectra to identify lithology, rock strength, presence of oil,presence of gas and the like. In some cases, a form of calibration usingindependently obtained elastic constants may be used. Some alternativeapproaches are configured to solve the following homogeneous system ofequations for known elastic constants C_(ij),

$\begin{matrix}{{\begin{bmatrix}{- a_{1}} & 0 & 0 & {d_{l}C_{11}} & {d_{2}C_{12}} & {d_{3}C_{13}} \\0 & {- a_{2}} & 0 & {d_{1}C_{12}} & {d_{2}C_{11}} & {d_{3}C_{13}} \\0 & 0 & {- a_{3}} & {d_{1}C_{13}} & {d_{2}C_{13}} & {d_{3}C_{33}}\end{bmatrix}\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} = 0} & (1)\end{matrix}$

The “a” terms and “d” terms refer to measured force (or acceleration)and displacement measurements derived from drilling data. This equationis a re-arrangement of the compressional stress-strain (σ−ε)relationship for a formation with vertical transversely isotropic (VTI)symmetry,

$\begin{matrix}{\begin{bmatrix}\sigma_{1} \\\sigma_{2} \\\sigma_{3}\end{bmatrix} = {{\begin{bmatrix}C_{11} & C_{12} & C_{13} \\C_{12} & C_{11} & C_{13} \\C_{13} & C_{13} & C_{33}\end{bmatrix}\begin{bmatrix}ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix}}.}} & (2)\end{matrix}$

In the case of an isotropic formation C33=C11 and C12=C13. Thecalibration vector is a set of scaling constants that map the drillingdata to stress and strain. Thus equation (2) is equivalent to equation(1) with

σ₁ =Aa ₁,ε₁ =Dd ₁,σ₂ =Ba ₂,ε₂ =Ed ₂,σ₃ =Ca ₃, and ε₃ =Fd ₃.  (3)

A scaling solution using singular value decomposition (SVD) may be foundfrom an instance of the data:

$\begin{matrix}{{\begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix} = {{\rho_{1}\begin{bmatrix}u_{1} \\u_{2} \\u_{3} \\u_{4} \\u_{5} \\u_{6}\end{bmatrix}} + {\rho_{2}\begin{bmatrix}v_{1} \\v_{2} \\v_{3} \\v_{4} \\v_{5} \\v_{6}\end{bmatrix}} + {\rho_{3}\begin{bmatrix}w_{1} \\w_{2} \\w_{3} \\w_{4} \\w_{5} \\w_{6}\end{bmatrix}}}},} & (4)\end{matrix}$

where the 6×1 column vectors u, v, and w are the zero-eigenvectors ofthe SVD decomposition of the 3×6 matrix of equation (1). The choice of(ρ1, ρ2, ρ3) may be arbitrary and the solution (ρ1=1, ρ2=ρ3=0) may beused. The scaling solution is fixed for all further processing. In someexamples, data from drilling cement with known properties may be used.However, one drawback to these alternative calibration techniques isthat the scaling solution may not work as well for other values of theelastic coefficients. As clarification, suppose the correct calibrationfor converting data to stress-strain is given by

$\begin{matrix}{S \equiv \begin{bmatrix}A \\B \\C \\D \\E \\F\end{bmatrix}} & (5)\end{matrix}$

In general the solution, S′, determined using the method describedabove, will be different, i.e. S′≠S due to the degeneracy of havingthree zero-eigenvectors. It is likely that new data, (a′, d′),satisfying the true scaling, S, but for different elastic constants,C_(ij)′, will not exactly satisfy equation (1), or equivalently equation(2), with the estimated S′. For example:

$\begin{matrix}{\begin{bmatrix}\sigma_{1} \\\sigma_{2} \\\sigma_{3}\end{bmatrix} = {\begin{bmatrix}{Aa}_{1}^{\prime} \\{Ba}_{2}^{\prime} \\{Ca}_{3}^{\prime}\end{bmatrix} = {{\begin{bmatrix}C_{11}^{\prime} & C_{12}^{\prime} & C_{13}^{\prime} \\C_{12}^{\prime} & C_{11}^{\prime} & C_{13}^{\prime} \\C_{13}^{\prime} & C_{13}^{\prime} & C_{33}^{\prime}\end{bmatrix}\begin{bmatrix}ɛ_{1} \\ɛ_{2} \\ɛ_{3}\end{bmatrix}} = {\begin{bmatrix}C_{11}^{\prime} & C_{12}^{\prime} & C_{13}^{\prime} \\C_{12}^{\prime} & C_{11}^{\prime} & C_{13}^{\prime} \\C_{13}^{\prime} & C_{13}^{\prime} & C_{33}^{\prime}\end{bmatrix}\begin{bmatrix}{Dd}_{1}^{\prime} \\{Ed}_{2}^{\prime} \\{Fd}_{3}^{\prime}\end{bmatrix}}}}} & (6)\end{matrix}$

but usually:

$\begin{matrix}{\begin{bmatrix}{A^{\prime}a_{1}^{\prime}} \\{B^{\prime}a_{2}^{\prime}} \\{C^{\prime}a_{3}^{\prime}}\end{bmatrix} \neq {\begin{bmatrix}C_{11}^{\prime} & C_{12}^{\prime} & C_{13}^{\prime} \\C_{12}^{\prime} & C_{11}^{\prime} & C_{13}^{\prime} \\C_{13}^{\prime} & C_{13}^{\prime} & C_{33}^{\prime}\end{bmatrix}{\quad{\begin{bmatrix}{D^{\prime}d_{1}^{\prime}} \\{E^{\prime}d_{2}^{\prime}} \\{F^{\prime}d_{3}^{\prime}}\end{bmatrix} = {\begin{bmatrix}{D^{\prime}d_{1}^{\prime}} & {E\prime d_{2}\prime} & {F^{\prime}d_{3}^{\prime}} & 0 \\{F^{\prime}d_{2}^{\prime}} & {D^{\prime}d_{1}^{\prime}} & {F^{\prime}d_{3}^{\prime}} & 0 \\0 & 0 & \left( {{D^{\prime}d_{1}^{\prime}} + {E^{\prime}d_{2}^{\prime}}} \right) & {F^{\prime}d_{3}^{\prime}}\end{bmatrix}\begin{bmatrix}C_{11}^{\prime} \\C_{12}^{\prime} \\C_{13}^{\prime} \\C_{33}^{\prime}\end{bmatrix}}}}}} & (7)\end{matrix}$

A specific numerical example is shown below:

-   -   A=0.412862808173015; B=0.543956657160610; C=0.365958060520887;    -   D=0.242715877845507; E=0.867051636244817; F=0.709008181859393;    -   a₁=2.605219220535070; a₂=1.749470948875160;        a₃=3.616150158277514;    -   d₁=1.013567297159670; d₂=0.852013230145349;        d₃=0.909328513511015;    -   C₁₁=0.317773277600477; C₁₂=0.569354480210020;        C₁₃=0.894678680404625;    -   C₃₃=0.686073057028121.

An SVD scaling solution (three orthonormal zero-valued eigenvectors ofthe matrix in equation (1)) are:

${u = \begin{bmatrix}{{0.0}82545581834373} \\{{0.2}64303913853697} \\{{0.2}11115955776180} \\{{0.9}33064438287170} \\{{- 0.0439}\mspace{11mu} 2992411\mspace{11mu} 1251} \\{{- 0.0788710151}\mspace{14mu} 41056}\end{bmatrix}};{v = \begin{bmatrix}{{0.1}55979022276073} \\{{0.1}07835077018896} \\{{0.1}81816295508710} \\{{- 0.0449524280}\mspace{14mu} 29207} \\{{0.9}62174470064409} \\{{- 0.0564317626}\mspace{14mu} 17977}\end{bmatrix}};$ $w = {\begin{bmatrix}{{0.2}53501119298623} \\{{0.3}72432622314350} \\{{0.1}18656788031531} \\{{- 0.0831774418}\mspace{14mu} 95659} \\{{- 0.0575872343}\mspace{14mu} 72243} \\{{0.8}79043181306523}\end{bmatrix}.}$

In practice the true solution S is an unknown linear combination of[u,v,w]. The prior art suggests any linear combination of [u,v,w] willsatisfy the stress strain relationship of equation (2) in all cases(cases other than the one used to solve for the eigenvectors. This isnot true. If we choose the estimated solution as suggested in the priorart, S′=u, then equation (6) is satisfied but equation (7) is not. Let:

-   -   A′=u₁; B′=u₂; C′=u₃; D′=u₄; E′=u₅; F′=u₆;    -   d′₁=0.109680746406904; d′₂=0.663145080030699;        d′₃=0.954077199822116;    -   C′₁₁=0.827014474276367; C′₁₂=0.945381650565510;        C′₁₃=0.715887955463262; C′₃₃=0.713520374298564.

The accelerations are chosen to satisfy the true scaling of equation(6),

-   -   a′₁=(C′₁₁Dd′₁+C′₁₂ Ed′₂+C′₁₃Fd′₃)/A=2.542864167863783;    -   a′₂=(C′₁₂Dd′₁+C′₁₁Ed′₂+C′₁₃Fd′₃)/B=1.810707215078647;    -   a′₃=(C′₁₃Dd′₁+C′₁₃Ed′₂+C′₃₃Fd′₃)/C=2.495749498699005;        but they do not satisfy equation (7) using the estimated        scaling,    -   A′a′₁=0.209902202262094≠(C′₁₁D′d′₁+C′₁₂E′d′₂+C′₁₃F′d′₃)=0.003225347546347    -   B′        a′₂=0.478577003788414≠(C′₁₂D′d′₁+C′₁₁E′d′₂+C′₁₃F′d′₃)=0.018787212438398    -   C′a′₃=0.526892540795763≠(C′₁₃D′d′₁+C′₁₃E′d′₂+C′₃₃F′d′₃)=−0.0012835033364027

The farther the estimated scaling is from the true scaling and/or thenew elastic constants from the ones used to estimate the scaling thegreater the misfit to the stress-strain relationship.

The alternative technique described above assumes equality in equation(7) and suggests using it to solve for the new Cij. Since we are solvingfor the Cij using instead of the unknown S, the inequality will producevariability in the estimates of the Cij. Regarding equation (7), thetechnique described above assumes that, in practice because there areonly three equations and four unknowns the solution is constrained bymaking an approximation where C12=C13. However, this assumption may ormay not be appropriate. Additionally, the technique described abovelimits the calibration to a pre-supposed 1-to-1 mapping of specificscaled drilling data parameters to specific components of stress andstrain.

Aspects of the subject technology propose a new, more stable method forcalibration and estimation of elastic constants based on multiple linearregression and minimization of L2 norms (e.g., the Euclidean normmeasuring the distance of the vector coordinate from the origin of thevector space). Various aspects of the subject technology also allow ageneral linear mapping of the drilling data to stress and strainincluding bias terms. Various aspects of the subject technology easilyincorporate data fusion by allowing an arbitrary number of drilling datavariables and data from other sensors in the calibration and estimation.Various aspects of the subject technology make it easy to optimize thecalibration over numerous variables such as type of stress and/or strainexcitation, type of drill bit, number of drilling parameters used in theinversion, and number of estimated acoustic properties. Additionaltechnical advantages over other techniques will be noted in the detaileddescription. Thus, various aspects of the subject technology are bothnovel and improve the estimation of mechanical rock properties fromdrilling data.

FIG. 1A is a schematic diagram of an example Logging While Drillingwellbore operating environment, in accordance with various aspects ofthe subject technology. In FIG. 1A, a drilling arrangement is shown thatexemplifies a Logging While Drilling (commonly abbreviated as LWD)configuration in a wellbore drilling scenario 50. Logging-While-Drillingtypically incorporates sensors that acquire formation data. The drillingarrangement of FIG. 1A also exemplifies what is referred to asMeasurement While Drilling (commonly abbreviated as MWD) which utilizessensors to acquire data from which the wellbore's path and position inthree-dimensional space can be determined. FIG. 1A shows a drillingplatform 2 equipped with a derrick 4 that supports a hoist 6 for raisingand lowering a drill string 8. The hoist 6 suspends a top drive 10suitable for rotating and lowering the drill string 8 through a wellhead 12. A drill bit 14 can be connected to the lower end of the drillstring 8. As the drill bit 14 rotates, it creates a wellbore 16 thatpasses through various subterranean formations 18. A pump 20 circulatesdrilling fluid through a supply pipe 22 to top drive 10, down throughthe interior of drill string 8 and out orifices in drill bit 14 into thewellbore. The drilling fluid returns to the surface via the annulusaround drill string 8, and into a retention pit 24. The drilling fluidtransports cuttings from the wellbore 16 into the retention pit 24 andthe drilling fluid's presence in the annulus aids in maintaining theintegrity of the wellbore 16. Various materials can be used for drillingfluid, including oil-based fluids and water-based fluids.

Logging tools 26 can be integrated into the bottom-hole assembly 25 nearthe drill bit 14. As the drill bit 14 extends the wellbore 16 throughthe formations 18, logging tools 26 collect measurements relating tovarious formation properties as well as the orientation of the tool andvarious other drilling conditions. The bottom-hole assembly 25 may alsoinclude a telemetry sub 28 to transfer measurement data to a surfacereceiver 32 and to receive commands from the surface. In at least somecases, the telemetry sub 28 communicates with a surface receiver 32using mud pulse telemetry. In some instances, the telemetry sub 28 doesnot communicate with the surface, but rather stores logging data forlater retrieval at the surface when the logging assembly is recovered.

Each of the logging tools 26 may include one or more tool componentsspaced apart from each other and communicatively coupled by one or morewires and/or other communication arrangement. The logging tools 26 mayalso include one or more computing devices communicatively coupled withone or more of the tool components. The one or more computing devicesmay be configured to control or monitor a performance of the tool,process logging data, and/or carry out one or more aspects of themethods and processes of the present disclosure.

In at least some instances, one or more of the logging tools 26 maycommunicate with a surface receiver 32 by a wire, such as wired drillpipe. In other cases, the one or more of the logging tools 26 maycommunicate with a surface receiver 32 by wireless signal transmission.In at least some cases, one or more of the logging tools 26 may receiveelectrical power from a wire that extends to the surface, includingwires extending through a wired drill pipe.

Collar 34 is a frequent component of a drill string 8 and generallyresembles a very thick-walled cylindrical pipe, typically with threadedends and a hollow core for the conveyance of drilling fluid. Multiplecollars 34 can be included in the drill string 8 and are constructed andintended to be heavy to apply weight on the drill bit 14 to assist thedrilling process. Because of the thickness of the collar's wall,pocket-type cutouts or other type recesses can be provided into thecollar's wall without negatively impacting the integrity (strength,rigidity and the like) of the collar as a component of the drill string8.

FIG. 1B is a schematic diagram of an example downhole environment, postdrilling, in accordance with various aspects of the subject technology.In FIG. 1B, an example system 80 is depicted for conducting downholemeasurements after at least a portion of a wellbore has been drilled andthe drill string removed from the well. A downhole tool is shown havinga tool body 46 in order to carry out logging and/or other operations.For example, instead of using the drill string 8 of FIG. 1A to lowertool body 46, which can contain sensors and/or other instrumentation fordetecting and logging nearby characteristics and conditions of thewellbore 16 and surrounding formations, a wireline conveyance 44 can beused. The tool body 46 can be lowered into the wellbore 16 by wirelineconveyance 44. The wireline conveyance 44 can be anchored in the drillrig 42 or by a portable means such as a truck 45. The wirelineconveyance 44 can include one or more wires, slicklines, cables, and/orthe like, as well as tubular conveyances such as coiled tubing, jointtubing, or other tubulars.

The illustrated wireline conveyance 44 provides power and support forthe tool, as well as enabling communication between data processors48A-N on the surface. In some examples, the wireline conveyance 44 caninclude electrical and/or fiber optic cabling for carrying outcommunications. The wireline conveyance 44 is sufficiently strong andflexible to tether the tool body 46 through the wellbore 16, while alsopermitting communication through the wireline conveyance 44 to one ormore of the processors 48A-N, which can include local and/or remoteprocessors. Moreover, power can be supplied via the wireline conveyance44 to meet power requirements of the tool. For slickline or coiledtubing configurations, power can be supplied downhole with a battery orvia a downhole generator.

FIG. 2 is a flow diagram showing a top-level workflow for estimatingelastic constants from drilling data while drilling using calibrationfrom an offset well, in accordance with various aspects of the subjecttechnology. The example workflow shown in FIG. 2 shows a number ofstages in a particular configuration. However, other workflows mayinclude additional stages, fewer stages, or alternative stages.Furthermore, the stages may be in different configurations in accordancewith other aspects of the subject technology. At a high level, theworkflow for estimating elastic constants from drilling data consists ofusing a calibrated linear mapping to determine stress and strain fromthe drilling data. The stress and strain are then inverted and used toestimate the elastic constants of the formation. The drilling datagenerally requires pre-processing before estimating the stress andstrain. The different stages of the workflow are discussed in furtherdetail below.

FIG. 3 is a diagram illustrating a workflow for pre-processing drillingdata, in accordance with various aspects of the subject technology. Thedrilling data used in the method may consist of surface data, downholedata, both, or other data. As an example surface data may consist of(but is not limited to) time and depth tagged variables such as weighton bit (WOB), rate of penetration (ROP), and torque on bit (TOB). Thesevariables may be measured downhole using stress and strain gaugestogether with downhole navigation data from some combination ofaccelerometers, gyros, and magnetometers. Some or all of the sensors maybe placed at or near the drill bit. The downhole data may be timetagged.

In some embodiments, a system may be configured to synchronize thedownhole and surface clocks so that downhole data can be associated withdepth. The surface data may be collected at a lower data rate than thedownhole data and may be up-sampled to match the downhole data rate. Thesystem is configured to collect and interpolate all the data variablesfrom different sources to the same rate and correctly associated withdepth/time tags. After synchronization, the data collected whiledrilling new rock may be extracted for further processing. Finally, thedata is parsed into bins. Each bin is associated with an output set ofelastic constants (Cij) for a given output depth. Data from othersensors may also be used in the calibration and estimation such asreal-time density and gamma ray measurements.

After pre-processing the drilling data, the data is linearly mapped tostress and strain using a previously computed calibration resultdetermined as follows. The system is configured to compute elasticconstants as a function of depth using an external source. For exampleacoustic waveforms from a wireline acoustic tool could be processed toestimate anisotropic compressional and shear velocities. Somecombination of these velocities, assumed acoustic relationships,anisotropy estimates, and bulk density estimates can be used to computeelastic constants versus depth. According to some embodiments, ahorizontal transverse isotropy (HTI) and vertical transverse isotropy(VTI) algorithms may be used to estimate the anisotropy. These elasticconstants are treated as ground truth for calibration.

FIG. 4 is a diagram illustrating a calibration workflow, in accordancewith various aspects of the subject technology. After acquiring groundtruth data it is interpolated to the drilling data depths. Then theallowed drilling data excitations are chosen. This determines which Cijvalues or related combinations (e.g. Young's modulus) can be estimatedfrom the drilling data. A linear map is proposed (with optionalconstraints) for transforming a chosen set of drilling data variablesinto stress and strain values. Since stress is related to strain throughthe known elastic constants, the system can estimate the optimal linearmap using multiple linear regression. The system may further optimizethe linear map by trying different combinations of drilling data,constraints on the linear map, and allowed excitations. The optimizedlinear map can be applied to other nearby wells as shown in FIG. 2 toinvert for the elastic constants in those wells. Over time a library oflinear maps can be developed and catalogued with respect to variousparameters such as regional lithology and drill bit type.

Linear Regression Formulas for Inversion of Elastic Constants fromStress and Strain

The stresses, {right arrow over (σ)}, and strains, {right arrow over(ε)}, are related through the elastic constant matrix, C,

{right arrow over (σ)}=c{right arrow over (ε)}.  (8)

In a vertical well with horizontal layers the elastic constant matrixhas VTI (Vertical Transverse Isotropy) symmetry,

$\begin{matrix}{{C = {\begin{bmatrix}C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\0 & 0 & 0 & C_{44} & 0 & 0 \\0 & 0 & 0 & 0 & C_{44} & 0 \\0 & 0 & 0 & 0 & 0 & C_{66}\end{bmatrix} \equiv \begin{bmatrix}C_{P} & 0_{3 \times 3} \\0_{3 \times 3} & C_{S}\end{bmatrix}}},} & (9)\end{matrix}$

where C_(P) and C_(S) are the 3×3 compressional and shear elasticconstant matrices respectively, and

C ₁₂ =C ₁₁−2C ₆₆.  (10)

We will focus on the compressional matrix since methods developed thereare easily applied to the diagonal shear matrix. Multiple linearregression is used to estimate the elastic constants by minimizing theL2 norm square difference for a depth bin,

$\begin{matrix}{{O = {\sum\limits_{k}{\left( {{C_{P}{\overset{\rightharpoonup}{ɛ}}_{P,k}} - {\overset{\rightharpoonup}{\sigma}}_{P,k}} \right)^{\prime}\left( {{C_{P}{\overset{\rightharpoonup}{ɛ}}_{P,k}} - {\overset{\rightharpoonup}{\sigma}}_{P,k}} \right)}}},} & (11)\end{matrix}$

with respect to the elastic constants, where

$\begin{matrix}{{{\overset{\rightharpoonup}{ɛ}}_{P,k} = \begin{bmatrix}ɛ_{1,k} \\ɛ_{2,k} \\ɛ_{3,k}\end{bmatrix}},{{\overset{\rightharpoonup}{\sigma}}_{P} = \begin{bmatrix}\sigma_{1,k} \\\sigma_{2,k} \\\sigma_{3,k}\end{bmatrix}},} & (12)\end{matrix}$

and the objective function sums over all data samples, k, in the depthbin. Minimization of equation (11) can be done in various ways. Oneembodiment uses a matrix equation derived by setting the partialderivatives of the elastic constants to zero. The choice of allowedexcitations determines which components or combination of components ofC_(P) can be estimated. The results can be optimized with respect toallowed excitation. This is an improvement over the other techniques,which do involve the selection of excitation.

The strain eigenvectors of C_(P) are:

$\begin{matrix}{{{\overset{\rightharpoonup}{u}}_{+} = \begin{bmatrix}a_{+} \\a_{+} \\b_{+}\end{bmatrix}},{{\overset{\rightharpoonup}{u}}_{-} = \begin{bmatrix}a_{-} \\a_{-} \\b_{-}\end{bmatrix}},{{\overset{\rightharpoonup}{u}}_{0} = \begin{bmatrix}a_{0} \\{- a_{0}} \\0\end{bmatrix}},{where}} & (13) \\{{\frac{b_{\pm}}{a_{\pm}} = {{- \frac{\left( {C_{11} + C_{12} - \lambda_{\pm}} \right)}{C_{13}}} = {- \frac{2C_{13}}{\left( {C_{33} - \lambda_{\pm}} \right)}}}},} & (14)\end{matrix}$

and the eigenvalues are:

$\begin{matrix}{{\lambda_{0} = {\left( {C_{11} - C_{12}} \right) = {2C_{66}}}},{\lambda_{\pm} = {\frac{\begin{matrix}{\left( {C_{11} + C_{12} + C_{33}} \right) \pm} \\\sqrt{\left( {C_{11} + C_{12} + C_{33}} \right)^{2} - {4\left\lbrack {{C_{33}\left( {C_{11} + C_{12}} \right)} - {2C_{13}^{2}}} \right\rbrack}}\end{matrix}}{2}.}}} & (15)\end{matrix}$

The following lists the regression formulas for different combinationsof excitations.

-   -   A) Excite +/−/0. This requires an estimate of all 6 stresses and        strains. Minimization of the objective function yields (sum over        k implied)

$\begin{matrix}{\begin{bmatrix}{2\left( {ɛ_{1,k}^{2} + ɛ_{2,k}^{2}} \right)} & {4ɛ_{1,k}ɛ_{2,k}} & {2\left( {{ɛ_{1,k}ɛ_{3,k}} + {ɛ_{2,k}ɛ_{3,k}}} \right)} & 0 \\{4ɛ_{1,k}ɛ_{2,k}} & {2\left( {ɛ_{1,k}^{2} + ɛ_{2,k}^{2}} \right)} & {2\left( {{ɛ_{1k}ɛ_{3,k}} + {ɛ_{2,k}ɛ_{3,k}}} \right)} & 0 \\{2\begin{pmatrix}{{ɛ_{1,k}ɛ_{3,k}} +} \\{ɛ_{2,k}ɛ_{3,k}}\end{pmatrix}} & {2\left( {{ɛ_{1,k}ɛ_{3,k}} + {ɛ_{2,k}ɛ_{3,k}}} \right)} & \begin{matrix}{{2\left( {ɛ_{1,k}^{2} + ɛ_{2,k}^{2} + {2ɛ_{3,k}^{2}}} \right)} +} \\{4ɛ_{1,k}ɛ_{2,k}}\end{matrix} & {2\begin{pmatrix}{{ɛ_{1,k}ɛ_{3,k}} +} \\{ɛ_{2,k}ɛ_{3,k}}\end{pmatrix}} \\0 & 0 & {2\left( {{ɛ_{1k}ɛ_{3,k}} + {ɛ_{2,k}ɛ_{3,k}}} \right)} & {2ɛ_{3,k}^{2}}\end{bmatrix}{\quad{\begin{bmatrix}C_{11} \\C_{12} \\C_{13} \\C_{33}\end{bmatrix} = {\begin{bmatrix}{2\left( {{\sigma_{1,k}ɛ_{1,k}} + {\sigma_{2,k}ɛ_{2,k}}} \right)} \\{2\left( {{\sigma_{1,k}ɛ_{2,k}} + {\sigma_{2,k}ɛ_{1,k}}} \right)} \\{2\begin{pmatrix}{{\sigma_{3,k}e_{1,k}} + {\sigma_{1,k}ɛ_{3,k}} +} \\{{\sigma_{3,k}ɛ_{2,k}} + {\sigma_{2,k}ɛ_{3,k}}}\end{pmatrix}} \\{2\sigma_{3,k}ɛ_{3,k}}\end{bmatrix}.}}}} & (16)\end{matrix}$

This example is a more robust method than equation (7) from othertechniques. It is a novel multiple linear regression that allowsestimation of all four compressional Cij without using the assumptionC₁₂=C₁₃ because the matrix is a 4×4 matrix instead of a 3×4 matrix. Thesummation over k not taught in the other techniques also reduces theeffect of noise.

-   -   B) Excite +/−. This requires an estimate of all 4 stresses and        strains (ε₁=ε₂, σ₁=σ₂). Minimization of the objective function        yields

$\begin{matrix}{\begin{bmatrix}{\sum\limits_{k}ɛ_{1,k}^{2}} & {\sum\limits_{k}{ɛ_{1,k}ɛ_{3,k}}} & 0 \\0 & {2{\sum\limits_{k}{ɛ_{1,k}ɛ_{3,k}}}} & {\sum\limits_{k}ɛ_{3,k}^{2}} \\{\sum\limits_{k}{ɛ_{1,k}ɛ_{3,k}}} & {\sum\limits_{k}\left( {{4ɛ_{1,k}^{2}} + ɛ_{3,k}^{2}} \right)} & {2{\sum\limits_{k}{ɛ_{1,k}ɛ_{3,k}}}}\end{bmatrix}{\quad{\begin{bmatrix}\left( {C_{11} + C_{12}} \right) \\C_{13} \\C_{33}\end{bmatrix} = {\begin{bmatrix}{\sum\limits_{k}{\sigma_{1,k}ɛ_{1,k}}} \\{\sum\limits_{k}{\sigma_{3,k}ɛ_{3,k}}} \\{{\sum\limits_{k}{\sigma_{1,k}ɛ_{3,k}}} + {2\sigma_{3,k}ɛ_{1,k}}}\end{bmatrix}.}}}} & (17)\end{matrix}$

Other techniques solve for Poisson's ratio and (C₁₁+C₁₂), oralternatively for Poisson's ratio and Young's modulus, using a straightline fit to drilling data. Thus these techniques result in two separate,different Poisson's ratios and do not resolve the discrepancy. Aspectsof the subject technology enable a system to directly solve for(C₁₁+C₁₂), C₁₃, and C₃₃, from which Poisson's ratio and Young's moduluscan be unambiguously determined using:

$\begin{matrix}{{v_{31} = \frac{C_{13}}{\left( {C_{11} + C_{12}} \right)}},{{{and}\mspace{14mu} E_{3}} = {C_{33} - {2v_{31}{C_{13}.}}}}} & (18)\end{matrix}$

Determining more of the elastic constants requires additionalinformation and/or assumptions. Shear elastic constants can be foundusing linear regression if the shear stresses and strains can beestimated form the drilling data. Alternatively, additional LWD sensorscould be added to the drill string (economically reasonable for acalibration run where acoustic data is required).

Example 1: C₄₄=C₅₅ known in real time from LWD Gamma Ray density and LWDacoustic tool measuring S-wave. Assume a known epsilon-gamma ratioEGR=β. Then

$\begin{matrix}{{C_{66} = {\frac{C_{33}c_{44}}{\left( {C_{44} - {\beta C_{33}}} \right)}\left\lbrack {1 - \beta - \frac{\left( {C_{11} + C_{12}} \right)}{2C_{33}}} \right\rbrack}},{C_{11} = {\frac{\left( {C_{11} + C_{12}} \right)}{2} + {C_{66}.}}}} & (19)\end{matrix}$

Example 2: Assume C₁₂=αC₁₃ with known α. Then we can get C₁₁ and C₆₆. Ifthe measurement of C₄₄ is known, the remaining unknown variables may beresolved.

Example 3: C₆₆ is known in real time from LWD Gamma Ray density and LWDacoustic tool measuring Stoneley wave velocity. The example may involvea tool correction to the Stoneley wave velocity. Then

$\begin{matrix}{C_{11} = {\frac{\left( {C_{11} + C_{12}} \right)}{2} + {C_{66}.}}} & (20)\end{matrix}$

Without an estimate of ε₁, one can still estimate Young's modulus andPoisson's ratio. Minimization of the objective function

$O = {\sum\limits_{k}\left( {\sigma_{3,k} - {2v_{31}\sigma_{1,k}} - {E_{3}ɛ_{3,k}}} \right)^{2}}$

reduced from equation (11) yields:

$\begin{matrix}{{\begin{bmatrix}{\sum\limits_{k}{ɛ_{3,k}\sigma_{3,k}}} \\{\sum\limits_{k}^{k}{\sigma_{3,k}\sigma_{1,k}}}\end{bmatrix} = {\begin{bmatrix}{\sum\limits_{k}ɛ_{3,k}^{2}} & {\sum\limits_{k}{ɛ_{3,k}\sigma_{1,k}}} \\{\sum\limits_{k}{ɛ_{3,k}\sigma_{1,k}}} & {\sum\limits_{k}\sigma_{1,k}^{2}}\end{bmatrix}\begin{bmatrix}E_{3} \\{2v_{31}}\end{bmatrix}}},} & (21)\end{matrix}$

where Young's modulus, E₃, and Poisson's ratio, v₃₁, are given by:

$\begin{matrix}{{{E_{3} = {{C_{33} - {2v_{31}C_{13}}} = {C_{33} - \frac{2C_{13}^{2}}{\left( {C_{11} + C_{12}} \right)}}}},{v_{31} = \frac{C_{13}}{\left( {C_{11} + C_{12}} \right)}}}.} & (22)\end{matrix}$

Other techniques only use estimation of Young's modulus and Poisson'sratio (or Poisson's ratio and C₁₁+C₁₂) from a straight line linearregression analysis applied to stress-stress, strain-strain, and orstress-strain ratios (e.g., slope and y-intercept directly give Young'smodulus and Poisson's ratio). Aspects of the subject technology differfrom these other techniques as aspects of the subject technology mayutilize a multiple linear regression (fits stress and strain to aplane).

-   -   C) Excite +/0. This requires an estimate of 5 stresses and        strains, ε₃=(ε₁+ε₂)b₊/2. Minimization of the objective function        yields

$\begin{matrix}{\begin{bmatrix}{\sum\limits_{k}\left( {ɛ_{1,\; k} + ɛ_{2,\; k}} \right)^{2}} & 0 & 0 \\0 & {\sum\limits_{k}\left( {ɛ_{1,\; k} - ɛ_{2,\; k}} \right)^{2}} & 0 \\0 & 0 & {\sum\limits_{k}\left( {ɛ_{1,\; k} + ɛ_{2,\; k}} \right)^{2}}\end{bmatrix} {\quad {\quad{\left\lbrack \begin{matrix}\lambda_{+} \\\lambda_{0} \\{\lambda_{+}b_{+}}\end{matrix} \right\rbrack = {\quad{\begin{bmatrix}{\sum\limits_{k}{\left( {\sigma_{1,\; k} + \sigma_{2,\; k}} \right)\left( {ɛ_{1,\; k} + ɛ_{2,\; k}} \right)}} \\{\sum\limits_{k}{\left( {\sigma_{1,\; k} - \sigma_{2,\; k}} \right)\left( {ɛ_{1,\; k} - ɛ_{2,\; k}} \right)}} \\{2{\sum\limits_{k}{\sigma_{3,\; k}\left( {ɛ_{1,\; k} + ɛ_{2,\; k}} \right)}}}\end{bmatrix}.}}}}}} & (23)\end{matrix}$

Converting from eigenvalues to elastic constants requires additionalinformation. Shear elastic constants can be found using linearregression if the shear stresses and strains can be estimated form thedrilling data. Alternatively, additional LWD sensors could be added tothe drill string.

Example 1: C₃₃ known in real time from LWD Gamma Ray density and LWDacoustic tool measuring P-wave. Then:

$\begin{matrix}{{C_{66} = {\lambda_{0}/2}},{C_{13} = {- \frac{b_{+}\left( {C_{33} - \lambda_{+}} \right)}{2}}},{C_{11} = \frac{\lambda_{+} + \lambda_{0} - {C_{13}b_{+}}}{2}},{C_{12} = {\frac{\lambda_{+} - \lambda_{0} - {C_{13}b_{+}}}{2}.}}} & (24)\end{matrix}$

Example 2: C₄₄=C₅₅ known in real time from LWD Gamma Ray density and LWDacoustic tool measuring S-wave. Assume a known epsilon-gamma ratioEGR=β. Then:

$\begin{matrix}{{C_{66} = {\lambda_{0}/2}},{C_{33} = \frac{{b_{+}^{2}\lambda_{+}} - {2\left( {\lambda_{+} + \lambda_{0}} \right)}}{\left( {b_{+}^{2} - {4\kappa}} \right)}},{C_{11} = {C_{33}\kappa}},{C_{13} = \frac{{b_{+}\left( {\lambda_{+} + \lambda_{0}} \right)} - {2\kappa \; b_{+}\lambda_{+}}}{\left( {b_{+}^{2} - {4\kappa}} \right)}},{C_{12} = \frac{\lambda_{+} - \lambda_{0} - {C_{13}b_{+}}}{2}},} & (25)\end{matrix}$

where κ≡1−β+βC₆₆/C₄₄.

Example 3: Assume C₁₂=αC₁₃ for known α.

$\begin{matrix}{{C_{66} = {\lambda_{0}/2}},{C_{11} = \frac{\lambda_{+} + \lambda_{0} - b_{+}}{2}},{C_{12} = {C_{11} - \lambda_{0}}},{C_{13} = {C_{12}/\alpha}},{C_{33} = {\frac{{\lambda_{+}b_{+}} - {2C_{13}}}{b_{+}}.}}} & (26)\end{matrix}$

-   -   D) Excite −/0. Same equations as +/0 with the substitution +→−.        Calibration Formulas for Estimating Compressional Stress and        Strain from Drilling Data

This section focuses on compressional stress and strain, with theunderstanding that similar methods can be applied to the shear stressand strain. Let n=1, . . . , N_(D) be output depth bin index and k=1, .. . , K_(n) be drilling data index. Define a generic log matrix:

$\begin{matrix}{{\left\lbrack {{\overset{\rightharpoonup}{L}}_{n\; 1}\mspace{11mu} \ldots \mspace{14mu} {\overset{\rightharpoonup}{L}}_{{nK}_{n}}} \right\rbrack = \begin{bmatrix}{ROP}_{n\; 1} & \ldots & {ROP}_{n,\; K_{n}} \\{WOB}_{n\; 1} & \ldots & {WOB}_{n,\; K_{n}} \\{TOB}_{n\; 1} & \ldots & {TOB}_{n,\; K_{n}} \\\vdots & \vdots & \vdots \\{GR}_{n} & \ldots & {GR}_{n}\end{bmatrix}},} & (27)\end{matrix}$

where the length N_(L) column vectors are

$\begin{matrix}{{\overset{\rightharpoonup}{L}}_{nk} = {\begin{bmatrix}{ROP}_{nk} \\{WOB}_{n\; k} \\{TOB}_{n\; k} \\\vdots \\{GR}_{n}\end{bmatrix} \equiv {\begin{bmatrix}L_{1,\; {nk}} \\L_{2,\; {nk}} \\L_{3,\; {nk}} \\\vdots \\L_{N_{L},\; n}\end{bmatrix}.}}} & (28)\end{matrix}$

Note the gamma ray, GR, is duplicated to length K_(n). The calibrationassumes a general linear mapping of stress and strain to the logs,

$\begin{matrix}{{{\overset{\rightharpoonup}{ɛ}}_{P,\; {nk}} = {{A_{ɛ}{\overset{\rightharpoonup}{L}}_{nk}} + {\overset{\rightharpoonup}{B}}_{ɛ}}},{{\overset{\rightharpoonup}{ɛ}}_{P,\; {nk}} = \begin{bmatrix}ɛ_{1,\; {nk}} \\ɛ_{2,\; {nk}} \\ɛ_{3,\; {nk}}\end{bmatrix}},{A_{ɛ} = {3 \times N_{L}}},{{\overset{\rightharpoonup}{B}}_{ɛ} = {3 \times 1}},} & (29) \\{{{\overset{\rightharpoonup}{\sigma}}_{P,\; {nk}} = {{A_{\sigma}{\overset{\rightharpoonup}{L}}_{nk}} + {\overset{\rightharpoonup}{B}}_{\sigma}}},{{\overset{\rightharpoonup}{\sigma}}_{P,\; {nk}} = \begin{bmatrix}\sigma_{1,\; {nk}} \\\sigma_{2,\; {nk}} \\\sigma_{3,\; {nk}}\end{bmatrix}},{A_{\sigma} = {3 \times N_{L}}},{{\overset{\rightharpoonup}{B}}_{\sigma} = {3 \times 1}},} & (30)\end{matrix}$

where the linear mapping, (A_(ε), {right arrow over (B)}_(ε), A_(σ),{right arrow over (B)}_(σ)), is a global constant, independent of depth.

Note the logs can be any external data that maps to stress and strainthrough the general linear map defined above. The linear map includes abias term. The general linear map is an important improvement over theconventional techniques which only allow a 1-to-1 scaling. The bias termmakes it possible to remove systemic biased error from the data. Aspectsof the subject technology allow for using more logs than number ofstresses and strains. This makes a data fusion approach easy toimplement and is an improvement over prior techniques. This alsofacilitates optimizing with respect to number and choice of logssimultaneously with excitation type.

It can be assumed that truth data exists for at least some of theelastic constants comprising the matrix C_(n) (subscript P has beendropped for convenience) consistent with the chosen multiple linearregression analysis from the previous section discussing excitations.

$\begin{matrix}{C_{n} = {\begin{bmatrix}C_{11,\; n} & C_{12,\; n} & C_{13,\; n} \\C_{12,\; n} & C_{11,\; n} & C_{13,\; n} \\C_{13,\; n} & C_{13,\; n} & C_{33,\; n}\end{bmatrix}.}} & (31)\end{matrix}$

These can be computed using wireline acoustic data, anisotropyestimation algorithms, a density log, and some assumed geo-mechanicsrelationships. The calibration is accomplished by minimizing the globalobjective function,

$\begin{matrix}{{O_{g} = {\sum\limits_{n = 1}^{Nd}\; {\sum\limits_{k = 1}^{K_{n}}\; {\left( {{\overset{\rightharpoonup}{\sigma}}_{P,\; {nk}} - {C_{P,\; n}{\overset{\rightharpoonup}{ɛ}}_{P,\; {nk}}}} \right)^{\prime}\left( {{\overset{\rightharpoonup}{\sigma}}_{P,\; {nk}} - {C_{P,\; n}{\overset{\rightharpoonup}{ɛ}}_{P,\; {nk}}}} \right)}}}},} & (32)\end{matrix}$

with respect to the linear map. As compared with other techniques,aspects of the subject technology are far more robust and stable withrespect to noise since many different C_(n) are used to estimate thecalibration. After substitution of the generalized linear map the globalL2 norm objective function takes the form

$\begin{matrix}{O_{g} = {\sum\limits_{n = 1}^{Nd}\; {\sum\limits_{k = 1}^{K_{n}}\; {\left( {{A_{\sigma}{\overset{\rightharpoonup}{L}}_{nk}} + {\overset{\rightharpoonup}{B}}_{\sigma} - {C_{P,\; n}A_{ɛ}{\overset{\rightharpoonup}{L}}_{nk}} - {C_{P,\; n}{\overset{\rightharpoonup}{B}}_{ɛ}}} \right)^{\prime}{\left( {{A_{\sigma}{\overset{\rightharpoonup}{L}}_{nk}} + {\overset{\rightharpoonup}{B}}_{\sigma} - {C_{P,\; n}A_{ɛ}{\overset{\rightharpoonup}{L}}_{nk}} - {C_{P,\; n}{\overset{\rightharpoonup}{B}}_{ɛ}}} \right).}}}}} & (33)\end{matrix}$

Minimization of equation (33) can be done in various ways. For example asystem may use a matrix equation derived by setting the partialderivatives of the linear map to zero. The linear map may also bepartially constrained. For example, the generalized linear map may beassumed to be diagonal, e.g., each stress and strain maps uniquely toone of the logs using a bias and scale factor. The form of thecalibration equations depends on the excitation choice from the previoussection and the constraints applied to the linear map. Calibrationequations for the regression of equation (16) with the mappingcompletely unconstrained are fairly complicated. The solution vector isdefined as

{right arrow over (x)}=[A _(ε,11) , . . . ,A _(ε,1N) _(L) ,A _(ε,21) , .. . ,A _(ε,2N) _(L) ,A _(ε,31) , . . . ,A _(ε,3N) _(L) ,A _(σ,12) , . .. ,A _(ε,1N) _(L) ,A _(σ,21) , . . . ,A _(σ,2N) _(L) ,A _(σ,31) , . . .,A _(σ,3N) _(L) ,B _(ε,1) B _(ε,2) B _(ε,3) ,B _(σ,1) ,B _(σ,2) ,B_(σ,3)]′  (34)

where without loss of generality we set A_(σ,11)=1. All the unknowns maybe normalized with respect to A_(σ,11). In this case the solution is

{right arrow over (x)}=M ⁻¹ {right arrow over (y)},  (35)

where M and {right arrow over (y)} are found from the partialderivatives of the objective function,

$\begin{matrix}{{\partial_{x_{i}}O_{g}} = {{{\sum\limits_{j}{M_{ij}x_{j}}} - y_{i}} = 0.}} & (36)\end{matrix}$

In the event that M is singular a pseudo-inverse may be used. Given goodstatistics for this example, there is always one zero eigenvectorcorresponding to A_(σ,11)=0, but it can be disregarded since it is trulydegenerate and satisfies the stress-strain relationship for allstiffness coefficients unlike other alternative techniques.

The matrix M is large, so only the first diagonal block of M is shown.The first diagonal block of M couples partial derivatives of theobjective function with respect to elements of A_(ε) to the elements ofA_(ε) in x. These terms arise from the part of the objective functionthat is quadratic in A_(ε),

$\begin{matrix}{\sum\limits_{n = 1}^{Nd}\; {\sum\limits_{k = 1}^{K_{n}}\; {\left\{ {\sum\limits_{i,\; {j = 1}}^{N_{L}}\; {{L_{i,\; {nk}}\left\lbrack {A_{ɛ}^{\prime}C_{n}^{2}A_{ɛ}} \right\rbrack}_{ij}L_{j,\; {nk}}}} \right\}.}}} & (37)\end{matrix}$

Partial derivatives of equation (37) with respect to A_(ε,pq) give

$\begin{matrix}{{M_{rs} = {2{\sum\limits_{n = 1}^{Nd}\; {\sum\limits_{k = 1}^{K_{n}}\; {{L_{{q{(r)}},\; {nk}}\left\lbrack C_{n}^{2} \right\rbrack}_{{p{(r)}},\; {m{(s)}}}L_{{j{(s)}},\; {nk}}}}}}},{{for}{\mspace{11mu} \;}r},{s = 1},\ldots \mspace{14mu},{3N_{L}},} & (38) \\{where} & \; \\{{p = {\left\lfloor {\left( {r - 1} \right)/N_{L}} \right\rfloor + 1}},{m = {\left\lfloor {\left( {s - 1} \right)/N_{L}} \right\rfloor + 1}},{q = {{{rem}\left( {{r - 1},N_{L}} \right)} + 1}},{j = {{{rem}\left( {{s - 1},N_{L}} \right)} + 1.}}} & (39)\end{matrix}$

There is no contribution to y from equation (37). Other parts of thematrix M and vector y can be determined similarly. Calibration equationsare much simpler for the regression of equation (21) with a diagonalconstraint for the linear map,

σ_(3,nk) =L _(σ,nk) A _(σ) +B _(σ),ε_(3,nk) =L _(ε,nk) A _(ε) +B_(ε),σ_(1,nk) =L _(1,nk) A ₁ +B ₁.  (40)

This example results in

$\begin{matrix}{{\begin{bmatrix}{\sum\limits_{n,\; k}{E_{n}^{2}L_{ɛ,\; {nk}}^{2}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}L_{ɛ,\; {nk}}L_{1,\; {nk}}}} & {\sum\limits_{n,\; k}{E_{n}^{2}L_{ɛ,\; {nk}}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}L_{ɛ,\; {nk}}}} & {- {\sum\limits_{n,\; k}{E_{n}L_{ɛ,\; {nk}}}}} \\{\sum\limits_{n,\; k}{E_{n}v_{n}L_{ɛ,\; {nk}}L_{1,\; {nk}}}} & {\sum\limits_{n,\; k}{v_{n}^{2}L_{1,\; {nk}}^{2}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}L_{1,\; {nk}}}} & {\sum\limits_{n,\; k}{v_{n}^{2}L_{1,\; {nk}}}} & {- {\sum\limits_{n,\; k}{v_{n}L_{1,\; {nk}}}}} \\{\sum\limits_{n,\; k}{E_{n}^{2}L_{ɛ,\; {nk}}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}L_{1,\; {nk}}}} & {\sum\limits_{n}{E_{n}^{2}K_{n}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}K_{n}}} & {- {\sum\limits_{n,\; k}{E_{n}K_{n}}}} \\{\sum\limits_{n,\; k}{E_{n}v_{n}L_{ɛ,\; {nk}}}} & {\sum\limits_{n,\; k}{v_{n}^{2}L_{1,\; {nk}}}} & {\sum\limits_{n,\; k}{E_{n}v_{n}K_{n}}} & {\sum\limits_{n}{v_{n}^{2}K_{n}}} & {- {\sum\limits_{n}{v_{n}K_{n}}}} \\{- {\sum\limits_{n,\; k}{E_{n}L_{ɛ,\; {nk}}}}} & {- {\sum\limits_{n,\; k}{v_{n}L_{1,\; {nk}}}}} & {- {\sum\limits_{n,\; k}{E_{n}K_{n}}}} & {- {\sum\limits_{n}{v_{n}K_{n}}}} & N_{T}\end{bmatrix}\begin{bmatrix}A_{ɛ} \\A_{1} \\B_{ɛ} \\B_{1} \\B_{\sigma}\end{bmatrix}} = \begin{bmatrix}{\sum\limits_{n,\; k}{E_{n}L_{ɛ,\; {nk}}L_{\sigma,\; {nk}}}} \\{\sum\limits_{n,\; k}{v_{n}L_{1,\; {nk}}L_{\sigma,\; {nk}}}} \\{\sum\limits_{n,k}{E_{n}L_{\sigma,\; {nk}}}} \\{\sum\limits_{n,\; k}{v_{n}L_{3,\; {nk}}}} \\{- {\sum\limits_{n,\; k}L_{\sigma,\; {nk}}}}\end{bmatrix}} & (41)\end{matrix}$

where V=2v₃₁, E=E₃, and without loss of generality A_(σ) is set to 1.

Other calibration formulas can similarly be determined for the otherregression formulas of intermediate difficulty compared to these twoextreme examples. One can optimize the choice of excitation/regressionand calibration equations for an oil field depending on the availabilityand quality of the drilling data, sensor data, and truth data. Over timea library of linear maps can also be developed and catalogued withrespect to various parameters such as regional lithology and drill bittype.

According to some embodiments, the minimizations discussed above may bedone in various ways other than the matrix equations above. Minimizationalgorithms such as synthetic annealing or conjugate gradient may also beused.

According to some embodiments the calibrations described above may beinterpreted as neural networks jointly training each other through thestress-strain relationship.

FIG. 5 is a diagram illustrating an adaptive linear element (ADALINE)neural network representation of the stress and strain, in accordancewith various aspects of the subject technology. In the diagram of FIG.5, the output transfer function is the identity function. FIG. 6 is adiagram illustrating stress and strain networks training against eachother using the stress-strain relationship for known stiffness matrix C,in accordance with various aspects of the subject technology. FIG. 6helps illustrate how the calibration method can be interpreted as thestress and strain ADELINE networks training against each other throughthe stress strain relationship given known stiffness matrix C. Thetraining set is indexed by depth bin {n, k}. The optimal networkparameters, A_(σ), {right arrow over (B)}_(σ),A_(ε), {right arrow over(B)}_(ε) are determined by minimizing the L2 norm cost function (alsoknown as the Least Mean Square Error (LMSE) algorithm) over the trainingset. It should be clear to those skilled in the art that theoptimization can be made non-linear by adding hidden layers to thenetworks and using different minimization algorithms. It should also beunderstood that different transfer functions and cost functions may beemployed without deviating from the spirit of the invention. AlthoughFIGS. 5 and 6 illustrate the use of ADALINE neural networks, other typesof neural networks and/or machine learning techniques may be used aswell.

Estimating Pseudo-Stress-Strain Logs

As discussed above, the logs may map to the stresses and strains throughthe general linear mapping specified in equations (29) and (30). Thus werefer to the logs as pseudo-stress-strains and there may not be anyconstraints for the linear map per se. Various aspects of the subjecttechnology optimize the linear map (including the best or improvedconstraints). One simple and constrained example is given by equation(40) where the equations assume the mapping is diagonal. However, thecalibration equations may be derived without this constraint.Furthermore, some aspects of the subject technology are able to processthe drilling data in different ways to produce multiple logs.

Aspects of the subject technology provide for the computing of thepseudo-stress, L_(σ,nk) and pseudo-strain, L_(ε,nk) of equation (40)from the axial acceleration, G_(z). Alternative techniques involveestimating stress and strain from the axial acceleration. Thesetechniques use the root mean square (RMS) value of the acceleration togive the stress up to an overall scale factor,

$\begin{matrix}{{{R\; M\; S} = \left\{ {\frac{1}{T}{\int_{0}^{T}{{G_{z}^{2}(t)}{dt}}}} \right\}^{1/2}},} & (42)\end{matrix}$

These alternative techniques estimate strain from the axial displacementspectra Zero Frequency Level (ZFL). The strain calculation is atechnique taken from much larger length scale seismic processing. Theacceleration is Fourier transformed to the frequency domain and dividedby the square of the angular frequency to convert to the displacementspectrum. Then the strain is interpreted as the displacement at the‘corner turn’ frequency. The corner turn frequency is the low frequencypoint at which the higher frequency linear trend of the displacementspectrum on a log-log plot breaks.

However, these alternative techniques are associated with shortcomingswith respect to computing stress and strain. For example, theacceleration does not equal the stress. The axial acceleration of thedrill bit is proportional to the total axial force acting on the drillbit. This is roughly equal to the weight on bit minus the force of therock (i.e. the rock stress) pushing back against the bit. Thus thechanges in acceleration are roughly proportional to the stress as itloads and unloads during rock fracturing.

Various embodiments of the subject technology relate to estimating anaxial pseudo-stress using the standard deviation of the acceleration. Asan example, if the n'th depth bin has N·K_(n) total points, thepseudo-stress would be

L _(σ,nk) =std{G _(z,n)(Nk+1), . . . ,G _(z,n)((k+1)N)} for k=0, . . . K_(n)−1.  (43)

Accordingly, the pseudo-stress may also be computed as

L _(σ,nk)=mean{abs{G _(z,n)(Nk+1)− G _(z,nk) , . . . ,G _(z,n)((k+1)N)−G _(z,nk)}} for k=0, . . . K _(n)−1,  (44)

where

G _(z,nk)=mean{G _(z,n)(Nk+1), . . . ,G _(z,n)((k+1)N)}.  (45)

A further advantage of the subject technology over alternativetechniques is that it is much less susceptible to bias errors in theestimation of G_(z). Another drawback of alternative techniques is thelack of an obvious corner turn frequency in the displacement spectra.The data may be noisy by nature and the data points at low frequency aresparse on a log-log plot, so when one looks at a large number of spectrait becomes clear that a consistent, well defined corner turn often doesnot exist. Furthermore, computing the displacement spectrum requirestaking a fairly large Fourier transform of the accelerometer data anddividing by the square of the angular frequency. This is mathematicallyequivalent to integrating the accelerometer data twice over the lengthof the Fast Fourier transform (FFT), but results in displacement errorsdue to accumulated accelerometer errors over the length of the FFT.These errors can be significant and not well correlated to the rockstrain. The drift in the displacement should be removed beforeestimating the pseudo-strain.

Consequently, aspects of the subject technology include estimating anaxial pseudo-strain associated with the axial pseudo-stress

L _(ε,nk)=mean{abs{I _(z,n)(Nk+1)−(m _(nk)(0)+b _(nk)), . . . ,I_(z,n)((k+1)N)−(m _(nk)(N−1)+b _(nk))}} for k=0, . . . K _(n)−1,  (46)

where I_(z,n) is the numerical double integral of the axialacceleration, and m_(nk) and b_(nk) are the slope and intercept of thebest fit line to I_(z,nk). The length of the time segment, N, should belarge enough to see the fluctuations in I_(z,n) after removal of thedrift, but not so large that drift errors cannot be removed with alinear fit. The higher frequency components left after linear driftremoval may be better correlated to the pseudo-stress calculated fromthe accelerometer variations.

The calculations of pseudo-stress and pseudo-strain may be computed inthe time domain, but equivalent equations may also be expressed in thefrequency domain. Furthermore, these embodiments can also be applied toany ‘acceleration’, such as rotary acceleration of the drill bit. Oncecalculated, the pseudo-stress and pseudo-strain (and possibly otherlogs) are mapped to actual stress and strain using the calibrated linearmap.

FIG. 7 is a diagram illustrating an example process for identifyingproperties of a rock formation based on stress and strain metrics, inaccordance with various aspects of the subject technology. The exampleprocess shown in FIG. 7 includes a number of operations in a particularconfiguration. However, other processes in accordance with variousaspects of the subject technology may include additional operations,fewer operations, and/or alternative operations. Furthermore, theoperations may be in different configurations or orders (e.g., someoperations may be performed in parallel).

At operation 705, a system may be configured to receive sensor dataobtained from one or more sensors. The sensor data may be generated fromdrilling operations performed in a wellbore at or near a rock formation.The sensors may include sensors positioned on components of abottom-hole assembly and/or sensor or other sensor data generatorspositioned at a surface site. The sensors may be configured to collectsensor data including, for example, depth, time, acceleration, rate ofpenetration (ROP), weight on bit (WOB), torque on bit (TOB),magnetometer and gyroscope data, and the like. Other sensors generateacoustic data, density measurements, gamma ray measurements, and thelike.

In some cases, the sensor data may need some additional preprocessingbefore stress and strain metrics may be generated based on the sensordata. At operation 710, one or more preprocessing actions may be takenon the sensor data. The preprocessing of operation 710 may include, forexample, synchronizing sensor data from the one or more sensors. Forexample, the sensor data collected may consist of surface data, downholedata, other data from another source, or a combination. The differentsources of data and/or measurements in the data may be time and/or depthtagged. The different sources and/or measurements of data may also besampled at different rates. The system may be configured to synchronizethe different data elements so that they may all be associated with adepth and/or time. The system is configured to collect and interpolateall the data variables from different sources to the same rate andcorrectly associated with depth/time tags. The preprocessing ofoperation 710 may also include parsing the sensor data into bins,wherein each bin is associated with an output set of elastic constantsfor a given output depth.

At operation 715, the system generates stress metrics and strain metricsfor the sensor data based on a calibrated stress and strain map. Thestress and strain map may represent a relationship between statisticalproperties of recorded data to acoustic properties of rocks in aformation.

The stress and strain map may be generated by collecting drilling datafrom various sources and relating the sensor data in the collecteddrilling data and other variables (e.g., drill bit type, rock formationcharacteristics, etc.) to the acoustic properties in the drilling datato build a library of prior calculations. The library may include datafrom offset wells, other remote wells, laboratory measurements in acontrolled environment, or earlier recorded data from the same well.

The stress and strain map may be calibrated by using various regressiontechniques (e.g., multiple linear regressions) to estimate theparameters of a linear map and also optimizing the structure of thelinear map. Non-linear maps and regressions may also be performedinstead of or in combination with other regression techniques by addinghidden layers or polynomial inputs to the neural networks representingthe stress and strain.

Once an optimal mapping is determined with regards to type of stressand/or strain excitation, number of drilling parameters used in theinversion, and number of estimated acoustic elastic constants, multiplelinear regression is used to invert for the acoustic elastic constantsas a function of depth. The multiple linear regression performed by thesystem is configured to model the relationship between two or moreexplanatory variables and a response variable by fitting a multiplelinear equation to observed data. The resulting estimates are obtainedcheaply in real-time or near real-time without the need of additionalexpensive sonic measurements.

At operation 720, the system may identify one or more properties of arock formation based on the stress metrics and the strain metrics. Insome embodiments, the stress metrics and strain metrics may includevector components of the stress and strain determined by applying theoptimal linear map to the logs. The properties of the rock formation mayinclude one or more elastics constants of the rock formation. Theelastic constants of the rock formation may be calculated based on themapping of logs to stress metrics and strain metrics and then usingmultiple linear regression on the stress and strain to estimate theelastic constants.

FIG. 8, which illustrates an example computing device architecture 800which can be employed to perform various steps, methods, and techniquesdisclosed herein. The various implementations will be apparent to thoseof ordinary skill in the art when practicing the present technology.Persons of ordinary skill in the art will also readily appreciate thatother system implementations or examples are possible.

As noted above, FIG. 8 illustrates an example computing devicearchitecture 800 of a computing device which can implement the varioustechnologies and techniques described herein. For example, the computingdevice architecture 800 can implement the various training systems,detection systems, data processors, downhole tools, servers, or othercomputing devices and perform various steps, methods, and techniquesdisclosed herein. The components of the computing device architecture800 are shown in electrical communication with each other using aconnection 805, such as a bus. The example computing device architecture800 includes a processing unit (CPU or processor) 810 and a computingdevice connection 805 that couples various computing device componentsincluding the computing device memory 815, such as read only memory(ROM) 820 and random access memory (RAM) 825, to the processor 810.

The computing device architecture 800 can include a cache of high-speedmemory connected directly with, in close proximity to, or integrated aspart of the processor 810. The computing device architecture 800 cancopy data from the memory 815 and/or the storage device 830 to the cache812 for quick access by the processor 810. In this way, the cache canprovide a performance boost that avoids processor 810 delays whilewaiting for data. These and other modules can control or be configuredto control the processor 810 to perform various actions. Other computingdevice memory 815 may be available for use as well. The memory 815 caninclude multiple different types of memory with different performancecharacteristics. The processor 810 can include any general purposeprocessor and a hardware or software service, such as service 1 832,service 2 834, and service 3 836 stored in storage device 830,configured to control the processor 810 as well as a special-purposeprocessor where software instructions are incorporated into theprocessor design. The processor 810 may be a self-contained system,containing multiple cores or processors, a bus, memory controller,cache, etc. A multi-core processor may be symmetric or asymmetric.

To enable user interaction with the computing device architecture 800,an input device 845 can represent any number of input mechanisms, suchas a microphone for speech, a touch-sensitive screen for gesture orgraphical input, keyboard, mouse, motion input, speech and so forth. Anoutput device 835 can also be one or more of a number of outputmechanisms known to those of skill in the art, such as a display,projector, television, speaker device, etc. In some instances,multimodal computing devices can enable a user to provide multiple typesof input to communicate with the computing device architecture 800. Thecommunications interface 840 can generally govern and manage the userinput and computing device output. There is no restriction on operatingon any particular hardware arrangement and therefore the basic featureshere may easily be substituted for improved hardware or firmwarearrangements as they are developed.

Storage device 830 is a non-volatile memory and can be a hard disk orother types of computer readable media which can store data that areaccessible by a computer, such as magnetic cassettes, flash memorycards, solid state memory devices, digital versatile disks, cartridges,random access memories (RAMs) 825, read only memory (ROM) 820, andhybrids thereof. The storage device 830 can include services 832, 834,836 for controlling the processor 810. Other hardware or softwaremodules are contemplated. The storage device 830 can be connected to thecomputing device connection 805. In one aspect, a hardware module thatperforms a particular function can include the software component storedin a computer-readable medium in connection with the necessary hardwarecomponents, such as the processor 810, connection 805, output device835, and so forth, to carry out the function.

For clarity of explanation, in some instances the present technology maybe presented as including individual functional blocks includingfunctional blocks comprising devices, device components, steps orroutines in a method embodied in software, or combinations of hardwareand software.

In some embodiments the computer-readable storage devices, mediums, andmemories can include a cable or wireless signal containing a bit streamand the like. However, when mentioned, non-transitory computer-readablestorage media expressly exclude media such as energy, carrier signals,electromagnetic waves, and signals per se.

Methods according to the above-described examples can be implementedusing computer-executable instructions that are stored or otherwiseavailable from computer readable media. Such instructions can include,for example, instructions and data which cause or otherwise configure ageneral purpose computer, special purpose computer, or a processingdevice to perform a certain function or group of functions. Portions ofcomputer resources used can be accessible over a network. The computerexecutable instructions may be, for example, binaries, intermediateformat instructions such as assembly language, firmware, source code,etc. Examples of computer-readable media that may be used to storeinstructions, information used, and/or information created duringmethods according to described examples include magnetic or opticaldisks, flash memory, USB devices provided with non-volatile memory,networked storage devices, and so on.

Devices implementing methods according to these disclosures can includehardware, firmware and/or software, and can take any of a variety ofform factors. Typical examples of such form factors include laptops,smart phones, small form factor personal computers, personal digitalassistants, rackmount devices, standalone devices, and so on.Functionality described herein also can be embodied in peripherals oradd-in cards. Such functionality can also be implemented on a circuitboard among different chips or different processes executing in a singledevice, by way of further example.

The instructions, media for conveying such instructions, computingresources for executing them, and other structures for supporting suchcomputing resources are example means for providing the functionsdescribed in the disclosure.

In the foregoing description, aspects of the application are describedwith reference to specific embodiments thereof, but those skilled in theart will recognize that the application is not limited thereto. Thus,while illustrative embodiments of the application have been described indetail herein, it is to be understood that the disclosed concepts may beotherwise variously embodied and employed, and that the appended claimsare intended to be construed to include such variations, except aslimited by the prior art. Various features and aspects of theabove-described subject matter may be used individually or jointly.Further, embodiments can be utilized in any number of environments andapplications beyond those described herein without departing from thebroader spirit and scope of the specification. The specification anddrawings are, accordingly, to be regarded as illustrative rather thanrestrictive. For the purposes of illustration, methods were described ina particular order. It should be appreciated that in alternateembodiments, the methods may be performed in a different order than thatdescribed.

Where components are described as being “configured to” perform certainoperations, such configuration can be accomplished, for example, bydesigning electronic circuits or other hardware to perform theoperation, by programming programmable electronic circuits (e.g.,microprocessors, or other suitable electronic circuits) to perform theoperation, or any combination thereof.

The various illustrative logical blocks, modules, circuits, andalgorithm steps described in connection with the examples disclosedherein may be implemented as electronic hardware, computer software,firmware, or combinations thereof. To clearly illustrate thisinterchangeability of hardware and software, various illustrativecomponents, blocks, modules, circuits, and steps have been describedabove generally in terms of their functionality. Whether suchfunctionality is implemented as hardware or software depends upon theparticular application and design constraints imposed on the overallsystem. Skilled artisans may implement the described functionality invarying ways for each particular application, but such implementationdecisions should not be interpreted as causing a departure from thescope of the present application.

The techniques described herein may also be implemented in electronichardware, computer software, firmware, or any combination thereof. Suchtechniques may be implemented in any of a variety of devices such asgeneral purposes computers, wireless communication device handsets, orintegrated circuit devices having multiple uses including application inwireless communication device handsets and other devices. Any featuresdescribed as modules or components may be implemented together in anintegrated logic device or separately as discrete but interoperablelogic devices. If implemented in software, the techniques may berealized at least in part by a computer-readable data storage mediumcomprising program code including instructions that, when executed,performs one or more of the method, algorithms, and/or operationsdescribed above. The computer-readable data storage medium may form partof a computer program product, which may include packaging materials.

The computer-readable medium may include memory or data storage media,such as random access memory (RAM) such as synchronous dynamic randomaccess memory (SDRAM), read-only memory (ROM), non-volatile randomaccess memory (NVRAM), electrically erasable programmable read-onlymemory (EEPROM), FLASH memory, magnetic or optical data storage media,and the like. The techniques additionally, or alternatively, may berealized at least in part by a computer-readable communication mediumthat carries or communicates program code in the form of instructions ordata structures and that can be accessed, read, and/or executed by acomputer, such as propagated signals or waves.

Other embodiments of the disclosure may be practiced in networkcomputing environments with many types of computer systemconfigurations, including personal computers, hand-held devices,multi-processor systems, microprocessor-based or programmable consumerelectronics, network PCs, minicomputers, mainframe computers, and thelike. Embodiments may also be practiced in distributed computingenvironments where tasks are performed by local and remote processingdevices that are linked (either by hardwired links, wireless links, orby a combination thereof) through a communications network. In adistributed computing environment, program modules may be located inboth local and remote memory storage devices.

It will be appreciated that for simplicity and clarity of illustration,where appropriate, reference numerals have been repeated among thedifferent figures to indicate corresponding or analogous elements. Inaddition, numerous specific details are set forth in order to provide athorough understanding of the embodiments described herein. However, itwill be understood by those of ordinary skill in the art that theembodiments described herein can be practiced without these specificdetails. In other instances, methods, procedures, and components havenot been described in detail so as not to obscure the related relevantfeature being described. Also, the description is not to be consideredas limiting the scope of the embodiments described herein. The drawingsare not necessarily to scale and the proportions of certain parts havebeen exaggerated to better illustrate details and features of thepresent disclosure.

In the above description, terms such as “upper,” “upward,” “lower,”“downward,” “above,” “below,” “downhole,” “uphole,” “longitudinal,”“lateral,” and the like, as used herein, shall mean in relation to thebottom or furthest extent of the surrounding wellbore even though thewellbore or portions of it may be deviated or horizontal.Correspondingly, the transverse, axial, lateral, longitudinal, radial,etc., orientations shall mean orientations relative to the orientationof the wellbore or tool. Additionally, the illustrated embodiments areillustrated such that the orientation is such that the right-hand sideis downhole compared to the left-hand side.

The term “coupled” is defined as connected, whether directly orindirectly through intervening components, and is not necessarilylimited to physical connections. The connection can be such that theobjects are permanently connected or releasably connected. The term“outside” refers to a region that is beyond the outermost confines of aphysical object. The term “inside” indicate that at least a portion of aregion is partially contained within a boundary formed by the object.The term “substantially” is defined to be essentially conforming to theparticular dimension, shape or other word that substantially modifies,such that the component need not be exact. For example, substantiallycylindrical means that the object resembles a cylinder, but can have oneor more deviations from a true cylinder.

The term “radially” means substantially in a direction along a radius ofthe object, or having a directional component in a direction along aradius of the object, even if the object is not exactly circular orcylindrical. The term “axially” means substantially along a direction ofthe axis of the object. If not specified, the term axially is such thatit refers to the longer axis of the object.

The terms “proximal” and “distal” are used herein with reference to auser manipulating the tool. The term “proximal” referring to the portionclosest to the user and furthest from the collar and the term “distal”referring to the portion located away from the user and closest to thecollar. It will be further appreciated that, for convenience andclarity, spatial terms such as “vertical”, “horizontal”, “up”, and“down” may be used herein with respect to the drawings. Notwithstanding,tools are used in many orientations and positions, and these terms arenot intended to be limiting and/or absolute.

Although a variety of information was used to explain aspects within thescope of the appended claims, no limitation of the claims should beimplied based on particular features or arrangements, as one of ordinaryskill would be able to derive a wide variety of implementations. Furtherand although some subject matter may have been described in languagespecific to structural features and/or method steps, it is to beunderstood that the subject matter defined in the appended claims is notnecessarily limited to these described features or acts. Suchfunctionality can be distributed differently or performed in componentsother than those identified herein. The described features and steps aredisclosed as possible components of systems and methods within the scopeof the appended claims.

Moreover, claim language reciting “at least one of” a set indicates thatone member of the set or multiple members of the set satisfy the claim.For example, claim language reciting “at least one of A and B” means A,B, or A and B.

Statements of the disclosure include:

Statement 1. A method of comprising receiving sensor data obtained fromone or more sensors, the sensor data generated from drilling operationsperformed in a wellbore; generating stress metrics and strain metricsfor the sensor data based on a calibrated stress and strain map; andidentifying one or more properties of a rock formation based on thestress metrics and the strain metrics.

Statement 2. The method of Statement 1, wherein the one or moreproperties comprises one or more elastics constants of the rockformation.

Statement 3. The method of Statements 1 through 2, further comprisingcalculating the calibrated stress and strain map using multiple linearregression on drilling and sensor data logs with known elasticconstants.

Statement 4. The method of Statements 1 through 2, further comprisingcalculating unknown elastic constants using multiple linear regressionon the stress metrics and strain metrics derived from applying thecalibrated stress and strain map to drilling and sensor data logs.

Statement 5. The method of Statements 1 through 4, further comprisingpreprocessing the sensor data, wherein the preprocessing of the sensordata comprises synchronizing sensor data from the one or more sensors.

Statement 6. The method of Statements 1 through 5, further comprisingparsing the sensor data into a plurality of bins, wherein each bin isassociated with an output set of elastic constants for a given outputdepth.

Statement 7. The method of Statements 1 through 6, further comprisinggenerating the calibrated stress and strain map.

Statement 8. The method of Statements 1 through 7, further comprisingcalibrating the calibrated stress and strain map using multiple linearregression with known elastic constants.

Statement 9. The method of Statements 1 through 8, further comprisingcalibrating the stress and strain map using hidden layers, polynomialinputs, or other neural network optimization techniques in the neuralnetworks representing stress and strain.

Statement 10. The method of Statements 1 through 9, wherein the one ormore sensors include at least one sensor positioned on a component of abottom-hole assembly and/or at least one sensor positioned at a surfacesite.

Statement 11. The method of Statements 1 through 10, wherein the sensordata comprises (but is not limited to) at least one of depth, time,axial acceleration, angular acceleration, rate of penetration (ROP),weight on bit (WOB), or torque on bit (TOB).

Statement 12. A system comprising one or more processors and at leastone non-transitory computer-readable medium having stored thereininstructions which, when executed by the one or more processors, causethe system to receive sensor data obtained from one or more sensors, thesensor data generated from drilling operations performed in a wellbore;generate stress metrics and strain metrics for the sensor data based ona stress and strain map; and identify one or more properties of a rockformation based on the stress metrics and the strain metrics.

Statement 13. The system of Statement 12, wherein the one or moreproperties comprises one or more elastics constants of the rockformation.

Statement 14. The system of Statements 12 through 13, wherein theinstructions further cause the system to calculate the one or moreelastic constants of the rock formation based on a linear regression ofstress metrics and strain metrics derived from applying the calibratedmap to the drilling and sensor data logs.

Statement 15. The system of Statements 12 through 14, wherein theinstructions further cause the system to synchronize sensor data fromthe one or more sensors.

Statement 16. Statements 12 through 15, wherein the instructions furthercause the system to comprise generating the stress and strain map basedon drilling data obtained in prior drilling operations.

Statement 17. The system of Statements 12 through 16, wherein the priordrilling operations are from at least one of an offset well, a remotewell, or laboratory measurements.

Statement 18. The system of Statements 12 through 17, wherein the priordrilling operations are from the wellbore.

Statement 19. A non-transitory computer-readable medium comprisinginstructions that, when executed by one or more processors, cause acomputing device to receive sensor data obtained from one or moresensors, the sensor data generated from drilling operations performed ina wellbore and identify one or more properties of a rock formation basedon a stress and strain map.

Statement 20. The non-transitory computer-readable medium of Statement19, wherein the instructions further cause the computing device togenerate stress metrics and strain metrics for the sensor data based onthe stress and strain map, wherein the one or more properties of therock formation are based on the stress metrics and the strain metrics.

Statement 21. The non-transitory computer-readable medium of Statements19 through 20, wherein the one or more sensors include at least onesensor positioned on a component of a bottom-hole assembly and/or atleast one sensor positioned at a surface site.

Statement 22: A system comprising means for performing a methodaccording to any of Statements 1 through 21.

What is claimed is:
 1. A method of comprising: receiving sensor dataobtained from one or more sensors, the sensor data generated fromdrilling operations performed in a wellbore; generating stress metricsand strain metrics for the sensor data based on a calibrated stress andstrain map; and identifying one or more properties of a rock formationbased on the stress metrics and the strain metrics.
 2. The method ofclaim 1, wherein the one or more properties comprises one or moreelastics constants of the rock formation.
 3. The method of claim 2,further comprising calculating the calibrated stress and strain mapusing multiple linear regression on drilling and sensor data logs withknown elastic constants.
 4. The method of claim 2, further comprisingcalculating unknown elastic constants using multiple linear regressionon the stress metrics and strain metrics derived from applying thecalibrated stress and strain map to sensor data.
 5. The method of claim1, further comprising preprocessing the sensor data, wherein thepreprocessing of the sensor data comprises synchronizing sensor datafrom the one or more sensors.
 6. The method of claim 1, furthercomprising parsing the sensor data into a plurality of bins, whereineach bin is associated with an output set of elastic constants for agiven output depth.
 7. The method of claim 1, further comprisinggenerating the calibrated stress and strain map.
 8. The method of claim1, further comprising calibrating the calibrated stress and strain mapusing multiple linear regression with known elastic constants.
 9. Themethod of claim 8, further comprising calibrating the stress and strainmap using hidden layers, polynomial inputs, or other neural networkoptimization techniques in neural networks representing stress andstrain.
 10. The method of claim 1, wherein the one or more sensorsinclude at least one sensor positioned on a component of a bottom-holeassembly and/or at least one sensor positioned at a surface site. 11.The method of claim 1, wherein the sensor data comprises at least one ofdepth, time, axial acceleration, angular acceleration, rate ofpenetration (ROP), weight on bit (WOB), or torque on bit (TOB).
 12. Asystem comprising: one or more processors; and at least onenon-transitory computer-readable medium having stored thereininstructions which, when executed by the one or more processors, causethe system to: receive sensor data obtained from one or more sensors,the sensor data generated from drilling operations performed in awellbore; generate stress metrics and strain metrics for the sensor databased on a stress and strain map; and identify one or more properties ofa rock formation based on the stress metrics and the strain metrics. 13.The system of claim 12, wherein the one or more properties comprises oneor more elastics constants of the rock formation.
 14. The system ofclaim 13, wherein the instructions further cause the system to calculatethe one or more elastic constants of the rock formation based on alinear regression of stress metrics and strain metrics derived fromapplying a calibrated stress and strain map to the sensor data.
 15. Thesystem of claim 12, wherein the instructions further cause the system tosynchronize sensor data from the one or more sensors.
 16. The system ofclaim 12, wherein the instructions further cause the system tocomprising generate the stress and strain map based on drilling dataobtained in prior drilling operations.
 17. The system of claim 16,wherein the prior drilling operations are from at least one of an offsetwell, a remote well, or laboratory measurements.
 18. The system of claim16, wherein the prior drilling operations are from the wellbore.
 19. Anon-transitory computer-readable medium comprising instructions that,when executed by one or more processors, cause a computing device to:receive sensor data obtained from one or more sensors, the sensor datagenerated from drilling operations performed in a wellbore; and identifyone or more properties of a rock formation based on a stress and strainmap.
 20. The non-transitory computer-readable medium of claim 19,wherein the instructions further cause the computing device to generatestress metrics and strain metrics for the sensor data based on thestress and strain map, wherein the one or more properties of the rockformation are based on the stress metrics and the strain metrics.